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Full text of "Mathematical And Physical Papers - Iii"

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The first case gives ft = ft, 71-y2 = a1-o2 = 0 or 180, so that the two streams into which it was proposed to resolve the first are of the same nature. The second case gives /32 = 90 ft, ry1  ry2 = oL1  a.2=9Q, which shews that the two streams are identical in their nature, only the first and second principal planes of the first of these streams are accounted respectively the second and first of the second stream. The third case gives y32=ft = 45, so that the two streams are circularly polarized and of the same kind, which is a particular instance of the first case.
Hence, universally, a stream of elliptically polarized light may be resolved into two streams of elliptically polarized light in which the polarizations are of any kind that we please, but different from one another.
Substituting for g2, &c. their values in (7), and replacing yv % by !  a, 2  a, for which they had been temporarily written, we find
{sin (ft   ) cos (2  a )
- J - I cos 08, + ft ) sin (03 - a )}"1 G1
= {sin 03 -ft) cos (a a,)
-1 cos 08 + ft,) sin (OL - e^)}-' G,
- J - 1 cos (ft + ft) sin (a2 - a,)}'1 G
3. Among these various modes of resolution there is one which possesses several peculiar properties, any one of which might serve to define it. Let us in the first place examine under what circumstances the intensity of the stream made up of the two components is independent of any retardation which the phase of vibration of one component may have undergone relatively to the phase of vibration of the other previously to the recomposition.
For this purpose there is evidently no occasion to consider the manner in which al9 blt e1? a2, 62, e2 are made up of a, 6, e, but we may start with the components. Let plt pz be the retardations of phase which take place before recomposition, and resolve the disturbances along the axes of #, y. We shall have for the resolved parts